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Mullineux involution and crystal isomorphisms
Nicolas Jacon
To cite this version:
Nicolas Jacon. Mullineux involution and crystal isomorphisms. 2021. �hal-03278927�
Mullineux involution and crystal isomorphisms
Nicolas Jacon
Abstract
We develop a new approach for the computation of the Mullineux involution for the symmetric group and its Hecke algebra using the notion of crystal isomorphism and the Iwahori-Matsumoto involution for the affine Hecke algebra of typeA. As a consequence, we obtain several new elementary combinatorial algorithms for its computation, one of which is equivalent to Xu’s algorithm (and thus Mullineux’ original algorithm). We thus obtain a simple interpretation of these algorithms and a new elementary proof that they indeed compute the Mullineux involution.
1 Introduction
The Mullineux problem is a long standing problem in the representation theory of the symmetric groups which has been studied by various authors since the end of the 70’s. Let Sn be the symmetric group onn letters withn >1. It is known that the irreducible representations ofSn over the field of complex numbers are naturally labeled by the partitions ofn (the sequences of non increasing positive integers of total sum n.)
IrrC(Sn) ={ρλ| λpartition of n}.
The characters and the dimensions of these representations may also been easily computed thanks to the combinatorics of partitions. There are exactly two non isomorphic representations of Sn with dimension 1: the trivial representation which is labeled by the partition (n) and the sign representationε, labeled by the partition (1. . . . .1)
| {z }
ntimes
. As a consequence, ifλis a partition ofn, there exists another partitionµsuch that ρµ ≃ ε⊗ρλ. It is natural to ask how one can compute µ from λ. The result is that µ is the conjugate partition of λwhich is defined by interchanging rows and columns in the Young diagram of λ(the Young diagram ofλis the finite collection of boxes arranged in left-justified rows, withλk boxes in thekth row for allk≥1.)
Of course, all the above questions and problems arise when we replace C by an arbitrary field k and in particular by a field of characteristicp >0. In this case, the irreducible representations have first been constructed in [12]. They are labeled by a subset of partitions called the set of p-regular partitions the partitions ofnwhere the non zero parts are not repeatedpor more times.
Irrk(Sn) ={ρeλ| λ p-regular partition ofn}.
We also have two one-dimensional representations: the trivial representation and the sign representationε and they are non isomorphic if and only if p 6= 2. By contrast, we still not even know how to compute the dimensions of these representations in general. The other mentioned problem still makes sense in this context. Namely, if λ is a p-regular partition then there exists a unique p-regular partition µ such that e
ρµ≃ε⊗ρeλ. If we set mp(λ) :=µ, we thus obtain an involutionmpon the set ofp-regular partitions.
If p= 2 then it is clear thatmp = Id (because then ε is nothing but the trivial representation) but in general, it is difficult to describemp. In fact, this map may even be defined in the context of Hecke algebras
2010Mathematics Subject Classification: 20C08,05E10
of typeAat ap-root of unity. In this case,pdo not need to be a prime but just a positive integer (greater than 2). The associated involution that we obtain coincides withmp ifpis prime. A natural problem is thus to find an explicit description of this involution me on the set ofe-regular partitions for alle∈N>1. This is the main subject of the present paper.
In [22], Mullineux has first given a conjectural algorithm for computing this involution (which will be called the Mullineux involution in the sequel). Later, another equivalent algorithm has been given by Xu [23, 24]. In [20], Kleshchev gave another combinatorial recursive algorithm for computing the Mullineux involution but it was not clear at that time why this algorithm would be equivalent to the Mullineux (and the Xu’s) algorithm. Ford and Kleshchev gave a proof of this fact later in [9]. Another proof was given in [3] by Bessenrodt and Olsson. In [5], Brundan and Kujawa gave another proof using works by Serganova on the general linear supergroup. We also note that recently, Fayers [8] has given another way for computing the involution.
The aim of this paper is to present several elementary combinatorial (and recursive) algorithms for the computation of the involution using the Kleshchev result. These algorithms are based on the results of [18, 19] and on the following points:
1. Each simple module for the Hecke algebra of typeAlabeled by ane-regular partition of rankncan be seen as a simple module for the affine Hecke algebra of type A.
2. The Mullineux map at the level of Hecke algebra coincide with the so called Iwahori-Matsumoto involution for the affine Hecke algebra of typeA.
3. The Iwahori-Matsumoto involution may be computed using an analogue involution at the level of Ariki-Koike algebras associated to a multicharges∈Zl.
4. This later involution may be computed using the Mullineux involution for Hecke algebras of typeAon e-regular partitions with rank (strictly) less than n.
As a consequence, to compute the image of ane-regular partition of ranknunder the Mullineux involution, we are reduced to compute several images of e-regular partitions of rank strictly less than n under the Mullineux involution. This thus gives a recursive algorithm to solve our problem. In fact, depending on the multicharge, we choose for our Ariki-Koike algebras, we obtain several different algorithms. It turns out that for a particular choice of multicharge, our algorithm is equivalent to Xu’s algorithm. This thus gives a new elementary proof for the fact that the Mullineux and the Xu’s algorithm give an answer for the Mullineux problem. This also gives a new interpretation of these algorithms (another interpretation is also given in [5]).
The paper will be organized as follows. In section 2, we recall some basic facts on the representation theory of affine Hecke algebras of typeAand of Ariki-Koike algebras. We also recall several results coming from [18, 17] concerning the labelling of the simple modules for these algebras and the relations between them. Section 3 introduces the Mullineux and the Iwahori-Matsumoto involutions and shows how these two maps are related. In section 4, we study combinatorial properties of partitions and multipartitions which will be used in the following sections. Section 5 gives the algorithms we get for computing the Mullineux involution. The last section shows that Xu’s algorithm can be seen as one of our algorithm.
Acknowledgement: The author thanks C´edric Lecouvey for fruitful discussions on the subject of this paper. The author is supported by ANR project JCJC ANR-18-CE40-0001.
2 Hecke algebras
In this first section, we recall the definitions of the affine Hecke algebra of type A and of the Ariki-Koike algebras. We then give a brief overview of their representation theories. Finally, we explain the relations between the known parametrizations of the simple modules for these algebras. The main references for these parts are [1] and [10].
2.1 Affine Hecke algebra of type A
Letn∈Z>0. Let q∈C∗ be a primitive root of unity of order e >1. TheIwahori-Hecke algebraHn(q) of typeAis the unital associativeC-algebra generated byT0,T1, . . . ,Tn−1 and subject to the relations:
TiTi+1Ti = Ti+1TiTi+1 (i= 1, . . . , n−2), TiTj = TjTi (|i−j|>1),
(Ti−q)(Ti+ 1) = 0 (i= 1, . . . , n−1).
Theaffine Hecke algebraHbn(q) is the unital associativeC-algebra which is isomorphic to Hn(q)⊗CC[X1±1, . . . , Xn±1],
as a C-vector space and such that Hbn(q) and C[X1±1, . . . , Xn±1] are both subalgebras of Hbn(q) with the following additional relations:
TiXiTi=qXi+1, TiXj=XiTj, for all (i, j)∈ {1, . . . , n−1}2withi6=j.
We denote by Modn the category of finite dimensionalHbn(q)-modules such that for allj= 1, . . . , n, the eigenvalues of the Xj are power ofq. The simple objects Irr(Hbn(q)) in Modn can be naturally labeled by the set of aperiodic multisegments that we now define:
Definition 2.1.1. Let l ∈N>0 and leti ∈ Z/eZ. The segment of lengthl and headi is the sequence of consecutiveresidues (i.e elements ofZ/eZ, identified with {0,1, . . . , e−1}) [i, i+ 1, . . . , i+l−1] inZ/eZ. The residuei∈Z/eZis then called theheadof the segment and the residuei+l−1 thetailof the segment.
A multisegment is a formal sum of segments. A multisegment is said to be aperiodic if for everyl ∈ Z>0, there existsi∈Z/eZsuch that there is no segment with length land taili appearing in the multisegment.
We denote by Me the set of aperiodic multisegments. The length of a multisegment is the sum of the lengths of the the segments appearing in it and is denoted by|ψ|. We denote byMe(n) the set of aperiodic multisegments of lengthn.
Example 2.1.2. Fore= 3, the multisegment:
[0,1,2,0] + [0] + [1] + [1,2] + [2,0]
is an aperiodic multisegment of length 10 where as
[0,1,2,0] + [0] + [0,1] + [1,2] + [2,0]
is a multisegment of length 10 which is not aperiodic.
By the geometric realization ofHbn(q) by Chriss and Ginzburg [7], we know that one may naturally label the simple modules in Modn by the setMe(n) of aperiodic multisegments of length n. We thus have:
Irr(Hbn(q)) ={Lψ |ψ∈Me(n)}
2.2 Ariki-Koike algebras
As above, we fix a primitive root of unity q∈C∗ of ordere >1. LetPl:=Zl and let{zi |i= 1, . . . , l} be the canonical basis ofPl. LetSl be the symmetric group generated by the transpositionsσi:= (i, i+ 1) for i= 1, . . . , l−1. The extended affine symmetric groupSblis the semidirect productPl⋊Slwith the relations given byσizj =zjσi forj 6=i, i+ 1 andσiziσi =zi+1 for i= 1, . . . , l−1 andj = 1, . . . , l. This group is generated by theσi fori= 1, . . . , l−1 and byτ:=zlσl−1. . . σ1 (see [18,§5.1].)
It acts faithfully onZl as follows: for anys= (s1, . . . , sl)∈Zl:
σc.s = (s1, . . . , sc−1, sc+1, sc, sc+2, . . . , sl) forc= 1, . . . , l−1 and zi.s = (s1, s2, . . . , si+e, . . . , sl) fori= 1, . . . , l.
and we have
τ.s= (s2, . . . , sl, s1+e).
Letsbe an orbit with respect to the above action and lets:= (s1, . . . , sl)∈Zl be an element in this orbit.
The Ariki-Koike algebraHns(q) is the quotientHbn(q)/Iswhere Is:=hQ
1≤j≤l(X1−qsj)i. If l= 1, this is a Hecke algebra of typeA(of finite type), and ifl= 2 a Hecke algebra of typeB (of finite type). One can see that the above algebra is well defined and depends only on the orbit ofsmodulo the action of Sbl (and on q).
The representation theory of this algebra has been intensively studied in a number of works. We refer to [1, 10] and the references theirin. We will only recall what is needed for the results of the present paper.
The analogues of the multisegments in the context of Ariki-Koike algebras are the multipartitions that we now define. For this, let us give some additional combinatorial definitions.
A partitionis a nonincreasing sequenceλ= (λ1,· · ·, λm) of nonnegative integers. One can assume this sequence is infinite by adding parts equal to zero. The rank of the partition is by definition the number
|λ|=P
1≤i≤mλi. We say thatλis a partition ofn, wheren=|λ|. By convention, the unique partition of 0 is the empty partition∅.
More generally, forl ∈Z>0, an l-partitionλof nis a sequence ofl partitions (λ1, . . . , λl) such that the sum of the ranks of the λj isn. The number nis then called the rank of λand it is denoted by|λ|. The set of l-partitions is denoted by Πl and the set ofl-partitions of rank nis denoted by Πl(n). Let λbe an l-partition. Thenodesor theboxesofλare by definition the elements of the Young diagram ofλ:
[λ] :={(a, b, c)|a≥1, c∈ {1, . . . , l}, 1≤b≤λca} ⊂Z>0×Z>0× {1, . . . , l}.
Thecontentof a nodeγ= (a, b, c) ofλis the elementb−a+sc ofZand the residue is the content modulo eZ. If l = 1 (that is when we consider a partition instead of a multipartition), then the Young diagram is identified with a subset ofZ>0×Z>0 in an obvious way.
Since the works of Ariki and Lascoux-Leclerc-Thibon, it is known that the representation theory of these algebras is closely related to the representation theory of quantum groups. In particular, one can naturally label the simple modules by the crystal basis of a certain integrable representation for the quantum group of affine type A. We will not give the details of all the consequences of this fact but we summarize this below. Again, we refer to [10] for a complete study. For all choice of s ∈ s, we can define a certain subset ofl-partitions which are called Uglovl-partitions and which are denoted by Φe,s(n). These classes of multipartitions, which strongly depends on the choice ofs, can all be seen as non trivial generalizations of the set ofe-regular partitions:
• For alls∈Z, we define:
Ale[s] :={(s1, . . . , sl)∈Zl |s1=s≤s2≤. . . sl< s+e}.
This is a fundamental domain for the action ofSblonZl. Ifs∈ Ale[s], then thel-partitions in Φe,s(n) are known as FLOTW l-partitions and they have a non recursive definitions: we haveλ= (λ1, . . . , λl)∈ Φs,e(n) if and only if:
1. For allj= 1, . . . , l−1 andi∈Z>0, we have:
λji ≥λj+1i+sj+1−sj. 2. For alli∈Z>0, we have:
λli≥λ1i+e+s1−sl. 3. For allk∈Z>0, the set
{λji−i+sj+eZ|i∈Z>0, λji =k, j= 1, . . . , l}, is a proper subset ofZ/eZ.
• Ifssatisfies for alli= 1, . . . , l−1,si+1−si> n−1 (we say thatsisvery dominant, it is also sometimes referred as the “asymptotic case” in the literature) then the set Φe,s(n) is known as the set Kleshchev l-partitions. Ifs′′ satisfy the same property, then the associated set Φe,s”(n) is the same.
• Ifl= 1, the set Φe,(s)(n) is simply the set ofe-regular partitions
It turns out that each set Φe,s(n) withs∈sgives a natural labelling for the irreducible representations of the Ariki-Koike algebraHsn(q). As a consequence, there are several natural possibilities for the labelling of the simple modules of Hns(q), one for each choice of an element in the orbit s. For more details on these parametrizations, we refer to [10]. Thus, one can write:
Irr(Hsn(q)) ={Dsλ|λ∈Φe,s(n)}.
By [6], each of these labellings has an interpretation in terms of a cellular structure. Last, clearly, if sand s′ in the same orbit, there is a bijection:
Ψse→s′ : Φ(e,s)(n)→Φ(e,s′)(n), which is uniquely defined as follows. For allλ∈Φ(e,s)(n) then:
Dλs ≃DΨ
s→s′
e (λ)
s′ .
This bijection has been explicitly described in [18] in a combinatorial way using crystal isomorphisms (the coincidence of the crystal isomorphisms with these bijections is proved in [14, Prop. 3.7].) We recall this description subsection (a program in GAP3 is available for computing it in all cases [15]). In the next sections, the following particular case: s= (s1, s2) ands′ = (s1, s2+e) will be of particular interest.
Remark 2.2.1. Ifs′ands′′are both very dominant multicharges in the same orbit then Ψse→s′is the identity.
Example 2.2.2. Assume thate= 3. Take s= (0 + 3Z,1 + 3Z). Taken= 3, then, we have Φ3,(0,1)(3) ={(∅,(3)),((1),(1,1)),((1),(2)),((2),(1)),((2,1),∅),((3),∅)}
Φ3,(0,4)(3) ={(∅,(3)),((1),(1,1(),((1),(2)),((2),(1)),(∅,(2,1)),((1,1),(1))}
Φ3,(1,0)(3) ={((3),∅),((1),(1,1)),((1),(2)),((1,1),(1)),((2,1),∅),((2),(1))}= Φ3,(4,0)(3) So that :
Irr(Hns(q)) = {D(∅,3)(0,1), D((1),(1,1))
(0,1) , D((1),(2))(0,1) , D((2),(1))(0,1) , D(0,1)((2,1),∅), D(0,1)((3),∅)}
= {D(∅,(3))(0,4) , D((1),(1,1))
(0,4) , D(0,4)((1),(2)), D((2),(1))(0,4) , D(∅,(2,1))(0,4) , D((1),(1,1)) (0,4) }
= {D(3,∅)(1,0), D((1),(1,1))
(1,0) , D((1),(2))(1,0) , D((1,1),(1))
(1,0) , D((2,1),∅)(1,0) , D((2),(1))(1,0) }
2.3 Description of the crystal isomorphisms
First let us assume thatl= 2 and that (s1, s2)∈Z2. Letλ∈Φ(e,s)(n). We follow the presentation in [18].
We define the minimal integerd≥ |s1−s2| such thatλ1d+1+s1−s2 =λ2d+1= 0 ifs2 ≥s1, and otherwise the minimal integer d≥ |s1−s2| such that λ2d+1+s2−s1 =λ1d+1 = 0. To (λ1, λ2), we associate itss-symbol of lengthd. This is the following two-rows array.
• Ifs1≤s2 then:
S(λ1, λ2) =
s2−d+λ2d . . . s2−2 +λ22 s2+λ21−1 s2−d+λ1d+s1−s2 . . . s1+λ11−1
• ifs1> s2then:
S(λ1, λ2) =
s1−d+λ2d+s2−s1 . . . s2+λ21−1
s1−d+λ1d . . . s1−2 +λ12 s1+λ11−1
We will writeS(λ1, λ2) = LL2
1
where the top row (resp. the bottom row) corresponds to λ2 (resp. λ1).
Of course, it is easy to recover the 2-partition from the datum of its symbol. From this symbol, we define a new symbol LLee2
1
as follows.
• Suppose first s2≥s1.Considerx1= min{t∈L1}.We associate tox1 the integery1∈L2 such that y1=
max{z∈L2|z≤x1}if min{z∈L2} ≤x1,
max{z∈L2} otherwise. (1)
We repeat the same procedure to the lines L2− {y1} and L1− {x1}. By induction this yields a sequence{y1, ..., yd+s1−s2} ⊂L2.Then we defineLe2 as the line obtained by reordering the integers of {y1, ..., yd+s2−s1}andLe1as the line obtained by reordering the integers ofL2− {y1, ..., yd+s1−s2}+L1
(i.e. by reordering the set obtained by replacing inL2the entriesy1, ..., yd+s1−s2 by those ofL1). We obtain a “symbol” LLee2
1
.
• Now, suppose s2< s1.Considerx1= min{t∈L2}.We associate tox1 the integery1∈L1 such that y1=
min{z∈L1|x1≤z}if max{z∈L1} ≥x1,
min{z∈L1}otherwise. (2)
We repeat the same procedure to the linesL1−{y1}andL2−{x1}and obtain a sequence{y1, ..., yd+s1−s2} ⊂ L1.Then we define Le1 as the line obtained by reordering the integers of{y1, ..., yd+s2−s1} andLe2 as the line obtained by reordering the integers ofL1− {y1, ..., yd+s2−s1}+L2.We obtain a “symbol” LLee2
1
.
The new symbol LLee2
1
that we obtain is canonically associated to a bipartition (λ1, λ2) and the multicharge (s2, s1). The crystal isomorphisms in the casel= 2 are thus entirely determined from the following results proved in [18]:
1. We have Ψ(se1,s2)→σ1(s1,s2)(λ1, λ2) = (λ1, λ2).
2. We have Ψ(se1,s2)→τ.(s1,s2)(λ1, λ2) = (λ2, λ1).
3. For allσ=x1. . . . .xm∈Sb2 withxi∈ {σ1, τ} for alli= 1, . . . , m, we have:
Ψ(se1,s2)→σ.(s1,s2)= Ψxe2...xm.(s1,s2)→σ.(s1,s2)◦. . .◦Ψ(se1,s2)→xm.(s1,s2) In the general casel∈N>0and s∈Zl, now:
1. For all c= 1, . . . , l−1, we have Ψ(se1,s2)→σc(s1,s2)(λ) =µ, where µj =λj for all j6=c, c+ 1,µc =λc andµc+1=λc+1.
2. We have Ψse→τ.s(λ) = (λ2, . . . , λl, λ1).
3. For allσ=x1. . . . .xm∈Sb2 withxi∈ {σ1, . . . , σl−1, τ} for alli= 1, . . . , m, we have:
Ψse→σ.s= Ψxe2...xm.s→σ.s◦. . .◦Ψse→xm.s
Example 2.3.1. Assume that (s1, s2) ∈ Z2 with s1 ≤ s2. In the next sections, we will be particularly interested in the computation of Ψ(se1,s2)→(s1,s2+e). Letλ= (λ1, λ2)∈Φ(e,s)(n), we then write its symbol:
S(λ1, λ2) =
s2−d+λ2d . . . s2−2 +λ22 s2+λ21−1 s2−d+λ1d+s1−s2 . . . s1+λ11−1
.
We then perform the above algorithm to obtain a new symbol eLeL2
1
which must be of the form : yd+s1−s2 . . . y1
xd . . . x2 x1
We then consider the following symbol:
0 . . . e−1 xd+e . . . x2+e x1+e yd+s1−s2 . . . y1
By the discussion above, this is the (s1, s2+e)-symbol of the bipartition Ψ(se1,s2)→(s1,s2+e)(λ1, λ2) (more details and examples can be found in [13])
Example 2.3.2. We keep the example 2.2.2, one can check that the map Ψ(0,1)→(0,4)
e is given as follows Ψ(0,1)→(0,4)
e : Φ3,(0,1) → Φ3,(0,4)
(∅,(3)) 7→ (∅,(3)) ((1),(1,1)) 7→ ((1),(1,1))
((1),(2)) 7→ (∅,(2,1)) ((2),(1)) 7→ ((2),(1)) ((2,1),∅) 7→ ((1,1),(1))
((3),∅) 7→ ((1),(2)) More examples can be found in [18].
2.4 Aperiodic multisegments and multipartitions
Letsbe an orbit ofZl with respect to the action of the affine symmetric group (recall the definition of the action in§2.2). If V is a simple module for the Ariki-Koike algebra then it is also a simpleHbn(q)-module in the category Modn. Hence there exists a unique aperiodic multisegment ψ such that V ≃ Lψ (as a Hbn(q)-module). As a consequence, far anys∈swe have a well defined map:
χne,s: Φ(e,s)(n)→Me(n),
which is defined as follows. Letλ∈Φ(e,s)(n), then we have a uniqueχn(e,s)(λ)∈Me(n) such that:
Dλs ≃Lχne,s(λ). By [2], this map may be described as follows:
• Assume first thats∈ Ale[s] for all non zero partλci ofλ, we associate the segment [(1−i+sc) +eZ, . . . , λci−i+sc].
By [2], The multisegmentχne,s(λ) is just the formal sum of all the segments associated to the non zero part ofλ.
• As a consequence, in general, ifs′∈s. Let s∈ Ale[s]∩s, then χne,s′(λ) =χne,s(Ψse′→s(λ)).
Given an aperiodic multisegmentψ, It is now natural to try to find the multicharges ssuch that ψ as an antecedent for the map χne,s. This question has been completely solved in [17]. There always exist such multicharges (they are non unique in general) which are called admissible multicharges. By [2], χne,s is injective so that ifsis admissible forψthere exists a uniqueλsuch thatχne,s(λ) =ψ. This l-partition will be calledadmissible(with respect to ψ). By definition, we have the following proposition where we use the following notation. Forsandttwo multicharges, we denotes⊂tif and only if, for allj∈Z/eZ, the number of integers conguent toj in sis less or equal to the number of integers conguent toj in t.
Proposition 2.4.1. Assume thatλ∈Φ(e,s)(n)thentis admissible for the multisegmentχne,s(λ)if and only if s⊂t.
Proof. Sets= (s1, . . . , sl) andt= (t1, . . . , tm). Assume thatλ∈Φ(e,s)(n) then as aHbn(q)-module, we have thatQ
1≤j≤l(X1−qsj) acts as 0 onDλs ≃Lχne,s(λ). As a consequence, ass⊂t, we have thatQ
1≤j≤m(X1−qtj) acts as 0 onLχne,s(λ). This implies that it is a well-definedHtn(q)-module and the result follows.
Remark 2.4.2. One can also prove the above proposition combinatorially using the descriptions of the ad- missible multicharges.
3 The Mullineux and the Iwahori-Matsumoto involutions
The aim of this section is to introduce the Mullineux involution for the symmetric group and its analogues in the context of Ariki-Koike algebras and affine Hecke algebras.
3.1 Iwahori-Matsumoto involution for affine Hecke algebras of type A
We have an involution♯onHbn(q) which has been defined by Iwahori and Mastumoto in [11]:
Ti♯=−qTi−1, Xj♯=Xj−1
fori= 1, . . . , n−1 andj= 1, . . . , n. The Iwahori-Matsumoto involution naturally induces an involution on the set of aperiodic multisegments. We have an involution:
♯:Me(n)→Me(n), defined for allψ∈Me(n) by
L♯ψ =Lψ♯.
Remark 3.1.1. We have in fact two others well defined involutions onHbn(q) which are defined as follows:
• The Zelevinsky involutionτ defined in [21] :
Ti♯=−qTn−i−1, Xj♯=Xn+1−j−1 , fori= 1, . . . , n−1 andj= 1, . . . , n.
• The involution∇ :
Ti∇=−qTn−i, Xj∇=Xn+1−j, fori= 1, . . . , n−1 andj= 1, . . . , n.
We have for allx∈Hbn(q):
xτ = (x∇)♯= (x♯)∇.
These two involutions thus also induce involutions on the setMe(n) and they have been studied in [17].
3.2 Mullineux involution for Ariki-Koike algebras
Assume thats∈Zl. Then we have a well-defined algebra automorphism:
γ:Hsn(q)→ Hsn(q−1), which is defined on the generators as follows:
T07→T0−1, Ti7→ −qTi−1.
This map naturally induces bijections on the indexing sets of the simple modules of Ariki-Koike algebras.
Lets♯ be the orbit of (−s1, . . . ,−sl) modulo the action of the affine symmetric group. Letv∈s♯ then we have a map:
mse→v: Φ(e,s)(n)→Φ(e,v)(n),
defined as follows. Letλ∈Φ(e,s)(n), then there exists a unique µ∈Φ(e,v)(n) such that (Dλs)γ ≃Dµv,
and we set
mse→v(λ) =µ.
This map has been described in [18]. If l = 1 and e is prime then it coincides with the usual Mullineux involution of the symmetric group that we have defined in the introduction. Ifl= 1, then it corresponds to the Mullineux involution of the Hecke algebra of typeA of [4] which will simply be denoted byme(it does not depend ons). In this paper, we will give an algorithm for computingme.
Remark 3.2.1. Ifλis a partition andγa node of its Young diagram, theγ-hook ofλid by definition the set of all the nodes at the right and at the bottom ofγ (includingγ). The length of the hook is the number of nodes in it. We say thatλis ane-core if all the hooks have length strictly less thane. Ifλis ane-core then me(λ) can be easily described: it is just the conjugation ofλ(as in the semisimple case), see [22] (wheneis a prime but the results generalizes easily ifeis an integer).
More generally, it is a natural question to ask how one can describe all the maps mse→v in general. It turns out that by [16, Prop. 4.2], knowing the mapme, one can describe it quite easily in a particular case:
Proposition 3.2.2. Assume that s is very dominant. Lets♯ := (−s′1, . . . ,−s′l) be a very dominant multi- charge such thats′i≡si+eZfor alli= 1, . . . , l. Then for allλ∈Φ(e,s)(n), we have:
mse→s♯(λ) = (me(λ1), . . . , me(λl)).
As a consequence, this result, combining with the fact that we know how to compute the natural bijection between the various parametrizations of the simple modules of Ariki-Koike algebras permit to describe all the Mullineux involutions (assuming that we knowme). Indeed, letv1 ∈sand letv2 ∈s♯. Lets1∈sbe a very dominant multicharge. Then we have:
mve1→v2= Ψse♯1→v2◦ms1→s♯1◦Ψve1→s1 wheres♯1 is as in the above proposition.
Example 3.2.3. We keep the setting of example 2.2.2. Forn= 3, the multicharge (0,4) is very dominant, so the above result applies in this case. One can takes♯= (0,5) which is also very dominant. Using the fact thatm3(3) = (2,1), m3(1.1) = (2), we obtain
m(0,4)→(0,5)
e Φ3,(0,1)(3) → Φ3,(0,5)(3) (∅,(3)) 7→ (∅,(2,1)) ((1),(1,1)) 7→ ((1),(2)) (∅,(2,1)) 7→ (∅,(3)) ((2),(1)) 7→ ((1,1),(1)) ((1,1),(1)) 7→ ((2),(1))
((1),(2)) 7→ ((1),(1,1))
Now combining with our cristal isomorphism in Example 2.3.2, we for example obtain m(0,1)→(0,5)
e : Φ3,(0,1) → Φ3,(0,5)
(∅,(3)) 7→ (∅,(2,1)) ((1),(1,1)) 7→ ((1),(2)) ((1),(2)) 7→ (∅,(3)) ((2),(1)) 7→ ((1,1),(1)) ((1,1),(1)) 7→ ((2),(1))
((3),∅) 7→ ((1),(1,1))
3.3 Relations between the involutions
Now we put all the above results together to deduce relations between the various involutions we have defined. The following result is proved in [17].
Theorem 3.3.1. Let ψbe an aperiodic multisegment and let s∈ Ale[s] be an admissible multicharge for ψ.
Set st= (−sl, . . . ,−s1)∈ Ale[−sl]then we have:
Ψ♯=χne,st◦mse→st◦(χne,s)−1(ψ)
As a consequence, the Iwahori-Mastumoto involution may be computed as follows. Take an aperiodic multisegmentψ.
• Choose an admissible multichargesforψ and computeλ:= (χne,s)−1(ψ) using§2.4.
• Computeν :=ms→se t(λ) using the discussion in the last section.
• Computeψ♯:=χne,st(ν) using the algorithm described in [17].
Example 3.3.2. Takee= 3 and the multisegment [0]+[0,1,2]+[1,2,3]. One can see that (0,1) is admissible for this multisegment and we have (χ73,(0,1))−1(ψ) = ((3),(3,1)).
We need to compute mse→st((3),(3,1)). To do this, we first compute Ψ(0,1)→(0,7)
e ((3),(3,1)) as (0,7) is very dominant. We obtain the bipartition ((1),(3,3)). Now we have seen that
m(0,7)→(0,8)
e ((1),(3,3)) = (m3(1), m3(3,3)) = ((1),(6)).
Again, we compute Ψ(0,8)→(0,2)
e ((1),(6)) = ((1),(6)) and thus we get ψ♯:= [0] + [2,0,1,2,0,1].
Now, let us explain how one can deduce an algorithm for computing the Mullineux involution fore-regular partitions. This is based on the following elementary remark. Letλ∈Φe,(0) be ane-regular partition and consider the aperiodic multisegment ψ :=χne,(0)(λ) (recall that this is nothing but the formal sum of the segments given by the rows of the Young diagram ofλ). The above theorem shows that:
me(λ) = (χne,(0))−1(ψ♯).
So now we are reduced to compute (χne,(0))−1(ψ♯). Takes∈ Ale[0] such thatl >1 then by Proposition 2.4.1, this is an admissible multicharge. We have:
ψ♯=χne,st◦ms→se t◦(χne,s)−1(ψ)
Nowµ:= (χne,s)−1(ψ) is the admissiblel-partition and the main problem is thus to computemse→st(µ). We have already seen that this can be done in three steps:
1. Compute the crystal isomorphism Ψs→ve (µ) = (ν1, . . . , νl) where v is very dominant (recall that this means that v= (s1, s2+ke) withke > n−1)
2. By Proposition 3.2.2, mve→v♯(ν) can be computed by applying the Mullineux map component by component. As |ν| = |λ|, if we assume that at least two components of the l-partition (ν1, . . . , νl) are non empty, all of the components are of rank < n and we know how to compute the Mullineux involution by induction.
3. Apply again a crystal isomorphism Ψve♯→st.
In the next section, we will apply the above algorithm in the case wherel= 2 and in particular show that the condition for applying our induction in step 2 is always satisfied (except in the case wheres= (s1, s2) ands1=s2.)
4 Combinatorial properties
In this section, we will try to find simple combinatorial ways to compute several objects that we have already defined: this concerns the admissible multicharges and multipartitions and the crystal isomorphisms.
4.1 On admissible multipartitions
Ifλ andµare two partitions, we denote by λ⊔ν the partition obtained by concatenation (and reordering the parts if necessary).
Assume that we have ane-regular partitionλ= (λ1, . . . , λr) (that is λ∈Φe,(s)(n) for any s∈Z). Let s ∈ Ale[s]. By Proposition 2.4.1, s is an admissible multicharge. The aim of this subsection is to show that one can easily construct the associated admissiblel-partitionλ∈Φe,s(n) such thatχne,s(λ) =χne,(s)(λ) (recall thatχne,sis always injective). To do this, one can use the algorithm developed in [17] from the datum of the multisegmentχne,(s)(λ) or we can argue as follows. Let l′ ∈ {1, . . . , l} be minimal such thatsl′ =sl. We constructλby induction as follows.
Ifλ=∅then λ:=∅and we are done. Otherwise, set
s′:=
(sl, . . . , sl
| {z }
l−l′+2
, s2+e, . . . , sl′−1+e) ifl′6= 1
s ifl′= 1
Note that we haves′∈ Ale[sl]. We denotem:=λ1+. . .+λe+s−sl.
By induction, we have constructed thel-partitionν∈Φ(e,s′)(n−m) such that we have χn−me,s′ (ν) =χn−me,(s
l)(λe+s−sl+1, λe+s−sl+2, . . . , λr) We then defineλas follows
• If we havel′= 1 thenλ1= (λ1, . . . , λe)⊔νlandλj =νj−1 ifj6= 1.
• Otherwise,λ1= (λ1, . . . , λe+s−sl)⊔ν2+l−l′ andλj=νj+1−l′ forj >1 where the indices are understood modulol.
Proposition 4.1.1. With this construction, we have λ∈Φe,s(n)andχne,s(λ) =χne,(s)(λ).
Proof. We prove the proposition by induction. The result is trivial whenn= 0. Keeping the above notations, one can assume that ν ∈Φ(e,s′)(n−m). First one can perform exactly the same procedure as in §2.4 for the description of the map χne,s to associate toλ a multisegment (even if we have - not already - proved that λ is in Φe,s(n)). By construction, this multisegment is nothing butχne,(s)(λ). It is thus an aperiodic multisegment. This proves condition 3 of FLOTWl-partition for λ (see the definition in§2.2). Hence, we just need to show that thel-partition satisfies the two first points.
• If l′ = 1, by induction, we haveνj≥νj+1 for allj= 1, . . . , l−1. This implies thatλji ≥λj+1i for all j = 2, . . . , l−1 and that λli ≥λ1i+e for alli≥1 and we get thatλ1i ≥λ2i because (λ1, . . . , λe) are the greatest parts ofλand becauseνil≥νi+e1 for alli >0.
• Ifl′ 6= 1, by the property of FLOTWl-partitions, we have thatµ:= (νl−l′+3, . . . , νl, ν1, . . . , νl−l′+1, νl−l′+2) is in Φe,v(n−m) for v = (s2, . . . , sl′−1, sl, . . . , sl, sl) and we can thus conclude using the fact that λ1j =λj ifj = 1, . . . , e+s−sl andλ1j =µlj−(e+s−sl)otherwise.
In the case wherel= 2 (which is the case that we will mostly studied in the forthcoming sections), the multipartitionλ= (λ1, λ2) is easy to obtain. One can assume thats1= 0, then we have
λ1= (λ1, . . . , λe−s2, λ2e−s2+1, . . . , λ3e−s2, . . . , λ2ke−s2+1, . . . , λ3ke−s2, . . .) and
λ2= (λe−s2+1. . . , λ2e−s2, λ3e−s2+1, . . . , λ4e−s2+1, . . . λ3ke−s2+1, . . . , λ4ke−s2, . . .)
Example 4.1.2. Let us takee= 4, λ= (8,8,6,6,4,3,3,2,1,1), then the associated Young tableau (with the residues of each node marked in the associated box) is:
0 1 2 3 0 1 2 3
3 0 1 2 3 0 1 2
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3
3 0 1 2 3 0 1 2 0 1
Takes= (0,2,2). Following the algorithm, we first have l′= 2. Then s′ = (2,2,2). We havem=λ1+λ2
and we need to computeνsuch that
χn−m4,(2,2,2)(ν) =χn−m4,(2)(6,6,4,3,3,2,1,1)
We obtainν= ((6,6,4,3),(3,2,1,1),∅) and we haveλ= ((8,8),(6,6,4,3),(3,2,1,1)).
In the case where l= 2, we have:
• Ifs= (0,0), we haveλ= ((8,8,6,6,1,1),(4,3,3,2)).
• Ifs= (0,1), we haveλ= (8,8,6,2,1,1),(6,4,3,3)).
• Ifs= (0,2), we haveλ= (8,8,3,2,1,1),(6,6,4,3)).
• Ifs= (0,3), we haveλ= ((8,3,3,2,1),(8,6,6,4,1)).
Using this, we have thus constructed a map
θe,ns: Φe,(0)(n)→Φe,s(n)
which associates toλthel-partitionλconstructed above (we will sometimes omit the subscriptn).
4.2 Crystal isomorphisms
In this second subsection, we study in details the crystal isomorphisms restricted to the multipartitions in the image ofθe,s. in the case wherel= 2. The first aim is to implify the procedure to compute it, the second is to show certain crucial properties which will show that our algorithm run.
Letλbe ane-regular partition and assume thats= (0, s). We also assume thatλis non empty and that ris maximal such that λr6= 0. Let (λ1, λ2) :=θ(e,(0,s))(λ) and consider the associated symbol with length tewitht sufficiently large. It is thus of the following form :
αte . . . α(t−1)e+1 . . . α2e . . . αe+1 αe . . . αs+1 . . . α1
βte−s . . . α(t−1)e−s+1 . . . β2e−s . . . βe−s+1 βe−s . . . β1
By definition of the symbol, we here haveαj:=λ2j−j+sforj= 1, . . . , keandβj:=λ2j−jforj= 1, . . . , ke−s.
We denote (µ1, µ2) := Ψ(0,s)→(0,s+k.e)
e (λ1, λ2) (so that, as usual,ke > n−1 and thus so that the multicharge (0, s+ke) is very dominant)
Assume thatλ6=∅ and thatµ2=∅ then the algorithm for the computation of Ψ(0,s)→(0,s+k.e)
e easily shows
that that this can happen if and only if Ψ(0,s)→(0,s+k.e)
e is the identity. This thus implies that {βi |i= 1, . . . , ke−s} ⊂ {αi |i= 1, . . . , ke}
In this case, we also need to haver≤e−s. Now we have for alli= 1, . . . , ke,αi =−i+sand alsoβj≤αj
for allj= 1, . . . , ke−s. As a consequence, we have
λ21−1≤ −1 +s and thusλ22≤s. We conclude
Proposition 4.2.1. Under the above notations, assume that µ2=∅ thenλ=λ1 is ane-core.
Proof. The above discussion shows thatλhas at moste−snon empty rows and at mostscolumns. This implies that the hooks ofλhas at most lengthe−1 and thus thatλis ane-core.
Now let us see what we can say ifµ1=∅. Before this, we show below that the image of λ under a crystal isomorphism can be quite easily computed in the case whereλis in the image of θe,s which is the case we are interested in here.
Keeping, the above notations, for all i = 1, . . . , k−1, we have αie = λ2ie−s−ie+s and βie−s+1 = λ2ie−s+1−(ie−s+ 1). So we have αie+ie−s≥βie−s+1+ie−s+ 1 and thus αie> βie−s+1.
In addition αie+1 = λ2ie+1−s −(ie+ 1) +s and β(i+1)e−s = λ2ie−s−((i + 1)e−s). So we have β(i+1)e−s+ ((i+ 1)e−s)≥αie+1+ (ie+ 1)−s. Soβ(i+1)e−s+e > αie+1.
These calculations show that one can perform our crystal isomorphism step by steps in the “blocks” of the symbol separated by vertical lines below. First recall in Example 2.3.1 how the crystal isomorphisms Ψ(0,s
′)→(0,s′+e)
e can be described.
αke . . . α(k−1)e+1 . . . α2e . . . αe+1 αe . . . αs+1 . . . α1
βke−s . . . β(k−1)e−s+1 . . . β2e−s . . . βe−s+1 βe−s . . . β1
We see that all the calculations in the blocks are trivial except in the rightmost. After one step of the crystal isomorphism we get
0 . . . e−1 . . . β3e−s+e . . . β2e−s+1+e β2e−s+e . . . βe+1 . . . βe−s+1+e . . . α′1 αke . . . α(k−1)e+1 . . . α2e . . . αe+1 β′e−s . . . β1′
and we see that the properties above are always satisfy. In particular, with the notations above, we have.
βe−s′ +e > βe−s+1+e
Now, take the right end of our first symbol:
αe . . . αs+1 αs . . . α1
βe−s . . . β1
We already know that βe−s+e > α1. Assume that we have λ1j 6= 0 so thatβj >−j. Then we claim that this implies that we haveβj ≥αs+j−1. To do this, note that we have:
βj≥βj−1+ 1≥. . .≥βe−s+ (e−s−j)> α1−s−j.
Now we haveα1≥α2+ 1≥. . .≥αs+j−1+ (s+j−2). So βj > αs+j−1−2
The only problem may appear ifβj=αs+j−1−1 and this implies that all the inequalities above are in fact equalities. We thus have:
βj=βj−1+ 1 =. . .=βe−s+ (e−s−j), and
α1=α2+ 1≥. . .=αs+j−1+ (s+j−2) =βj+s−j−1 =βj−1+s−j =. . .=βe−s+e−1.
This case implies that we have an e-period in the sense of [19, Def. 2.2]. Such property is impossible for Uglovl-partitions by [19, Prop. 5.1].
This discussion implies that, under the notations above, if we haveβj>−jthen we must haveβj′ >−j so that the associated part of the partition is also non zero. By a direct induction, we thus deduce:
Proposition 4.2.2. Let 0< s < e and letλ be an e-regular partition and (λ1, λ2) :=θ(e,(0,s))(λ). Assume that (µ1, µ2) := Ψ(0,s)→(0,s+k.e)
e (λ1, λ2) for k >>0 (so that (0, s+k.e) is very dominant, see §2.2). Then
|µ1| 6= 0.
Remark 4.2.3. In the case wheres= 0, the above discussion also shows that if (λ1, λ2) :=θ(e,(0,0))(λ) then Ψ(0,s)→(0,k.e)
e (λ1, λ2) = (∅, λ) fork >>0. As a consequence, this choice of multicharge cannot be used to get our recursive algorithm to compute the Mullineux involution because then it would require the computation ofme(λ) ... to computeme(λ).
5 The algorithm
Let λ = (λ1, . . . , λr) be an e-regular partition of rank n. We can now present a recursive algorithm for computing me(λ). First by Remark 3.2.1, one can assume that λ is not an e-core. The algorithm now consists in the following steps:
1. Choose 0< s < eand consider the bipartition (λ1, λ2) :=θ(e,(0,s))(λ).
2. Compute (µ1, µ2) := Ψ(0,s)→(0,s+k.e)
e (λ1, λ2) fork >>0. By Propositions 4.2.1 and 4.2.2, we now that
|µ1|< nand|µ2|< n.
3. By induction, we knowme(µ1) andme(µ2) and we can thus compute:
(κ1, κ2) := Ψ(0,−s+ke)→((0,e−s)
e (me(µ1), me(µ2)).
4. We haveme(λ) =θ(e,(0,e−s))−1 (κ1, κ2).
Note that in principle, one can choose an arbitrary multicharge s instead of (0, s) (as soon as the second point at the end of subsection 3.3 is satisfied) but the complexity of the algorithm for the computation of the crystal isomorphism from sto a very dominant multicharge increases. However, It is not unreasonable to expect that some particular multicharge can lead to interesting fast new algorithms.
5.1 Steps 1 and 2
It follows from Section 4.2 that the first two steps can be both implemented by the process below. Let 0< s < eand sets= (0, s). We set λ[1] = (λ1, . . . , λe−s) and λ[2] = (λe−s+1, . . . , λr), we write the Young tableau ofλ[1] with the associated contents and just below, the Young tableau ofλ[2] with the associated contents with respect to the multicharge (0, s).
λ[1]
0 1 2 3 . . . λ1−1
1 0 1 2 . . . λ2−2
... ... ... ... ...
e−s−1 . . . λe−s−(e−s)
λ[2]
s s+ 1 . . . λe−s+1−1 +s
s−1 . . . λe−s+2−2 +s . . . .
For example, takeλ= (10,8,7,5,4,4,3,2,1,1). Takee= 4 ands= 1 λ[1]
0 1 2 3 4 5 6 7 8 9
1 0 1 2 3 4 5 6
2 1 0 1 2 3 4
λ[2]
1 2 3 4 5
0 1 2 3
1 0 1 2
2 1 0
3 1
4 5
Now, starting with the first part ofλ[1], consider the content of the rightmost box, sayc. Inλ[2], we consider the rightmost boxes and we take the one with the greatest content which is less thanc, sayc′. Then we remove the boxes of the first part of λ[1] with content greater than c′ into this part in λ[2] (in other words, we move the “truncated first row” containing the boxes grater thanc to the row inλ[2]).
It is clear that we still have a partition. Then, we do the same for the second part ofλ[1] and so on until we reach the last part ofλ[1]. If this is not possible we switch to the second part ofλ[1], and we continue this process until we reach the last part ofλ[1].
In our example, we must remove the boxes in bold in the first partition above, and add the boxes in bold
in the second partition below.
λ[1]
0 1 2 3 4 5
1 0 1 2 3
2 1 0 1 2
λ[2]
1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6
1 0 1 2 3 4
2 1 0
3 1
4 5
We then collect all the parts of λ[2] that are above the smallest part we have modified, in a partition µ. So here µ = (9,7,6). The new partition λ[2] is given by the remaining parts and we add e to the contents of all the boxes in it. We then move the step above and continue the process until we cannot do anything. The remaining parts of λ[2] are added to µ. Then the partition λ[1] is the first component of Ψ(0,s)→(0,s+k.e)
e (λ1, λ2) andµis the second.
λ[1]
0 1 2 3 4 5
1 0 1 2 3
2 1 0 1 2
λ[2]
2 3 4
1 2
0 1 It becomes :
λ[1]
0 1 2 3 4
1 0 1 2
2 1 0
λ[2]
2 3 4 5
1 2 3
0 1 2
1
We have nowµ= (9,7,6,4,3,3), and we the above process:
λ[1]
0 1 2 3 4
1 0 1 2
2 1 0
λ[2]
3
